ELi5: Avatar's Favor and its effect on BP recast timer, specifically with Beckoner's Horn +1 (Avatar's Favor +3).
Basically there are two ways to reduce BP recast from the base 60 seconds.
1. BP recast stats. These cap at -15 each, and the overall cap is -30. So for example you can't just do -30 all in the BP recast category, anything beyond -15 is wasted and the rest needs to come from BP recast II or III. There's no gear that gives III, that only comes from job point gifts which give you up to -10 sec.
So with -10 from the gifts, you only need -20 from gear to cap it. It still caps at -15 per category so you still need some from recast I and some from recast II. For example, the most common scenario people have is -13 recast I (Baayami Cuffs and Slops) and -7 recast II (Sancus Sachet +1) for -20 total. Add the -10 from gifts and you're capped.
2. Avatar's Favor. This is a totally separate category, and isn't included in the overall -30 sec cap. It goes beyond that. For the most part, each tier of favor you add will reduce recast by another 1 second.
However, Favor does not start out at its max value. It starts out at tier 1, and it builds slowly over time. Your summoning skill, and Beckoner's Horn, determine where Favor will max out, not where it starts. The only thing that affects the strength of Favor on a freshly-summoned avatar is the job point category for Avatar's Favor. Anyone with capped Favor job points, and -30 sec recast, will generally have a BP recast of about 26 seconds on a fresh avatar.
As time goes by with the same avatar out, its max favor tier available increases. If your skill/favor bonuses are high enough, your avatar will then gain the increased favor tier. This happens constantly, so if you aren't wearing Beckoner's Horn +1 and then you equip it, if your avatar has been out for long enough then your favor tier will immediately jump up from equipping the horn.
Do I have to wear it at the time of activating Avatar's Favor or just swap it in in my BP precast sets? I have my lua set up to do the latter.
You only need to swap it in for BP precast set. (Or actually midcast works too, since it's a JA. So if you're doing it in precast, make sure something
else isn't getting swapped in for midcast because that will take precedence over the precast gear.)
Favor seems to keep track of how long it's been active for a given avatar, and when you change your gear, it adjusts the favor tier appropriately.
Looking at the chart on BG Wiki is a little confusing. It's basically a combination of summoning magic skill and avatar's favor + effect to reach a tier level, but hard to tell exactly how it interacts with the recast of BPs.
Yes that's basically it, plus you want the -30 to your recast but that won't be as obvious from looking at a master SMN's equip set because with Baayami gear, all the recast stats come from gear that's also BiS summoning skill. So it will look like it's just a full skill build, but really you do need those recast stats. Particularly, Sancus Sachet and Baayami Cuffs/Slops. Those will get you -19 seconds, gifts will get you -10 seconds, so you only need another -1 second. Espiritus is one way to get it. Sancus Sachet +1 is another. Conveyance Cape augment is another. There are lots of ways, you just need to make sure it's -20 total if you have both BP recast gifts.
BP Recast I, II, and gifts, will get you up to -30 sec recast, for a 30 sec timer. Then there's Favor, which is a totally separate reduction, which can give you up to another -9 sec for a total of -39.
Edit: -9 sec tier is very hard to hit (requires the 670 skill tier with Beckoner's Horn +1 equipped). -8 sec tier is not actually at 575 like you would expect, it seems to skip the 575 skill tier. To hit -8 seconds you only need 512 skill with Beckoner's Horn +1 equipped. That number is MUCH easier to hit and will give you 22 second BP recast. Again though, only on an avatar that's been out for a minute or so. Every resummon resets it.